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Random Variables and Probability Distributions 49

and, for part (b),

Z 6
P 2 < X 6†ˆ f x† dx ‡ p 3†
X X
2
1 Z 3 1 Z 6 1
ˆ e x=3 dx ‡ e x=3 dx ‡
3 2 6 3 2e
e 2
2=3
ˆ e
2
These results are, of course, the same as those obtained earlier using the PDF.

3.3 TWO OR MORE RANDOM VARIABLES

In many cases it is more natural to describe the outcome of a random experi-
ment by two or more numerical numbers simultaneously. For example, the
characterization of both weight and height in a given population, the study of
temperature and pressure variations in a physical experiment, and the distribu-
tion of monthly temperature readings in a given region over a given year. In
these situations, two or more random variables are considered jointly and the
description of their joint behavior is our concern.
Let us first consider the case of two random variables X and Y . We proceed
analogously to the single random variable case in defining their joint prob-
ability distributions. We note that random variables X and Y can also be
considered as components of a two-dimensional random vector, say Z. Joint
probability distributions associated with two random variables are sometimes
called bivariate distributions.

As we shall see, extensions to cases involving more than two random vari-
ables, or multivariate distributions, are straightforward.

3.3.1 JOINT PROBABILITY DISTRIBUTION FUNCTION

The joint probability distribution function (JPDF) of random variables X and Y ,
denoted by F XY (x, y), is defined by

F XY x; y†ˆ P X x \ Y y†; 3:16†

for all x and y. It is the probability of the intersection of two events; random
variables X and Y thus induce a probability distribution over a two-dimensional
Euclidean plane.

TLFeBOOK